Marquise has $200$ meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width $x$ (in meters) is modeled by: $A(x)=-x^2+100x$ What is the maximum area possible?
Answer: The garden's area is modeled by a quadratic function, whose graph is a parabola. The maximum area is reached at the vertex. So in order to find the maximum area, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $A(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} A(x)&=0 \\\\ -x^2+100x&=0 \\\\ x^2-100x&=0 \\\\ x(x-100)&=0 \\\\ \swarrow &\searrow \\\\ x=0\text{ or }&x-100=0 \\\\ x={0}\text{ or }&x={100} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({100})}{2}=\dfrac{100}{2}={50}$ The vertex's $x$ -coordinate is ${50}$. Now let's find $A({50})$ : $\begin{aligned} A({50})&=-({50})^2+100({50}) \\\\ &=-2500+5000 \\\\ &=2500 \end{aligned}$ In conclusion, the maximum garden area is $2500$ square meters.